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The concept of monoidal category does not presume that values of such aggregate types can be taken apart; on the contrary, it provides a framework that unifies classical and quantum information theory. [1] In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the ...
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms. μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon diagram. and the unitor diagram commute.
A biclosed monoidal category is a monoidal category that is both left and right closed. A symmetric monoidal category is left closed if and only if it is right closed. Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is left or right closed.
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology . [ 1 ]
Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an -monoidal category. More generally, the center of a E k {\displaystyle E_{k}} -monoidal category is an algebra object in E k {\displaystyle E_{k}} -monoidal categories and therefore, by Dunn additivity , an E k ...
Download as PDF; Printable version; ... Pages in category "Monoidal categories" ... Center (category theory) Closed monoidal category;
If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field. Given a field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the ...
They are a prominent tool in applied category theory. When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in ...